First of all, if we are going to have a proper ship, we need to fashion our 5,000 tons of steel into a shape that is a little more recognizable and functional as a ship. To do this job we’re going to need to work from a good set of plans or blueprints; builder’s plans. To get these we’re going to need professional help.

Lucky us!! Our neighbor Joe is a naval architect. Let’s invite him to a barbecue on our cube and see if he’ll help us out. Good neighbor that he is Joe prepares a set of plans for us and we set about turning our 5,000 tons of steel into a proper ship’s hull. When we’ve finished our hull is well shaped up to the “main deck” which will be the first continuous deck open to the weather and sea and which we can seal off with hatches to keep the water out. But our hull is quite empty and now has a volume of 10,000 cubic meters.

So with 10,000 cubic meters of volume and 5,000 metric tons of weight we now have 5,000 tons of spare or reserve buoyancy. We can see this reflected in the amount of our watertight hull remaining above the waterline. That’s called “freeboard.” The depth of the hull which is underwater is called “draft.”

Time for some illustrations. These are taken from the Handbook of Damage Control, NAVPERS 1691 published online by the San Francisco Maritime National Park Association. The first illustration shows a hull cross section and demonstrates draft, freeboard, and reserve buoyancy.

If we want to use our ship to haul cargo and passengers we still have a lot of improvements to make before we can do that. We’re going to need engines and propellers and all of the associated propulsion machinery to make it go. And we’re going to need to build accommodations for a crew and passengers. And we’ll need rudders and steering gear and a we’ll need to build a navigation bridge up high so we can steer the ship and see where we’re going. We’ll need electronics equipment, radars, radios, interior communications and such. We’ll also need cargo holds and fittings to carry and secure the cargo, and lots more. Some of this can go inside our watertight hull, but we’ll also need to build a “superstructure”, or additional decks and compartments above the watertight hull.

Well, it turns out that we need a lot of stuff and it’s really expensive. Better set up a reserve account with the bank to pay for it all. But it’s not just expensive, when we add it all up it’s also really, really heavy. And heavy is something we need to care a lot about. To see why, just think about Archimedes again. Suppose he hops into his full to the brim bath tub after a big turkey dinner which caused him to put on ten pounds. What happens when he floats in his tub? According to his principle he will sink a little lower (his draft will increase) until he displace exactly an additional ten pounds of water.

That’s going to happen with our ship too. Remember, we have a 5,000 ton hull and a 10,000 cubic meter or 10,000 ton buoyancy force limit. So that means we have 5,000 tons of extra buoyant capacity. If we’ve added 2,000 tons to our 5,000 ton hull with all of these modifications so that it now weighs 7,000 tons, it’ll sink further into the water until it displaces the additional 2,000 tons of water. That means we would now have a deeper draft, less freeboard, and only 3,000 tons of spare buoyancy left.

We need to add all of these things to our hull and still leave room for the weight of the fuel, oil, water, stores, cargo, crew, and passengers. We also want to have some residual or reserve buoyant capacity; because we don’t like to live dangerously, do we? Just like we set up that reserve account of money with the bank to pay for things, we’ll want to set up a reserve account of buoyancy as well.

You may be thinking that it would be nice if we could just make some sort of mark on our hull that would tell us when the draft of our hull has increased to the point where we have reached our maximum load and are about to encroach on our minimum reserve buoyancy. Well, you’re too late. George Plimsoll thought of that back in the 19th century. In fact, he not only thought of it, he made it his life’s work. Plimsoll, a member of the British Parliament, was very concerned with the safety of seamen and the dangers of badly overloaded ships. Like Archimedes, he too had something named after him. But this time it wasn’t a principle, but a mark on ship’s hulls. The Plimsoll line or Plimsoll mark can be found on every ocean-going ship. It gives a ready visual representation from which to tell if the ship has been overloaded and is sitting too deeply in the water with insufficient freeboard. Here’s an illustration. There are lots of marks shown but that’s because different water conditions and temperatures can affect the permissible load line.

RIGHTING ARM AND MOMENT

We have all of these things, heavy things, to put into and onto our hull. So where are we going to put them? Does it make a difference? Should we just pile them up and put them on top of our hull? Our common sense tells us that is not a very good idea because we’ll make something that is very top heavy and it might tip over. Guess what!! Our common sense is right! To the extent possible we’ll want the really heavy things down low and the lighter things on top.

We intuitively know this as a general principle but ships in the water are complex things acted upon by many external and internal forces and the physics of stability quickly gets very complicated. To understand just the basics of stability, why our ship remains upright and how we can keep it that way we’re going to have to concern ourselves with things like the center of gravity (G), center of buoyancy (B), metacenter (M), distance from the center of gravity to the metacenter (GM) and more.

The kind of stability we’re concerned with here is called transverse stability or what we commonly think of as roll. That’s the rotation of a ship around its longitudinal axis. There is also longitudinal stability but we’re not going to worry about that since it is unlikely it was a factor in the capsize of the Sewol.

Remember when I said you could put away your calculator? It might be a good idea to see if you can find it. And while you’re at it you might check to see if it has trigonometry functions.

The first thing we need to do is figure out the center of gravity of our ship (G). That’s a single point through which we may consider all of the gravitational forces acting on our ship to be concentrated. Don’t go through our ship looking for “G” though because you won’t find it. It’s a mathematical concept.

I know what you’re thinking now. You’re thinking “I sure wish there was some simple experiment I could do to demonstrate this.” Well, that’s easy — there is. Head for your lab (the kitchen) and pick up a fork and run your finger along the handle until you find the place where it balances perfectly. Bingo! You’ve discovered its center of gravity (well — actually you’ve discovered a vertical line within the fork handle on which it rests).

You can also demonstrate a related concept. Put a nice piece of steak on the fork and watch what happens (it’ll also work with tofu too if you’re so inclined). You’ll have to move your finger further along the fork toward the steak, but you’ll find a new place where it balances. You’ve just discovered another concept important to our discussion of the Sewol capsize. The center of gravity (G) always moves in the direction of added weight. Now bite off half the steak and leave the rest on the fork and watch what happens. Now you have to move your finger further back along the handle for it to balance. That’s the flip side of the same rule. The center of gravity (G) will always move away from a reduction of weight.

Unfortunately we can’t run our ship along our finger to find G. To do this with our ship we need to figure out the center of mass of the hull and every thing we’ve put into and onto it. At this point you might wish we had left this thing as a cube because that would make the math really simple. But our friend Joe, the naval architect, is up to the task. He does all of these calculations and comes up with our center of gravity (G).

Next, we want to look at the opposing force, the buoyancy (B). The buoyancy forces are going to be acting all along the entire underwater hull. So once again this is a job for our naval architect. He will look at the shape of our hull, the underwater profile, the volume that is displaced and arrive at a theoretical point through which all of these upward buoyant forces can be thought of as acting. That is the center of buoyancy of our ship (B).

If you were hoping for another lab experiment to demonstrate this (and I know you were) I won’t let you down. You can demonstrate this concept as well but it’s a little trickier. Get back to the lab (kitchen) and fill your sink with water. Not all the way to the top like Archimedes, you’re not going to overflow it. Now take something that will float and has sides like a roasting pan or even a pie tin. Put it in the water and push down on it so that you can feel the buoyancy force pushing back against your finger. You’ll cause the thing to tip until you find that sweet spot where you can push down and the whole object submerges uniformly around your finger. You’ve done it! You’ve discovered the center of buoyancy of your pan or whatever (or at least — the vertical line on which it rests).

Now try this to see another important and related concept that we’ll talk about later. Put something near the edge of your floating pan that’s heavy enough to tilt it but not sink it. Now once again find the spot where you can push down and the pan will submerge uniformly around you finger. You’re not trying to straighten it up — to the contrary, you want to find the spot where you can push down on it without changing the amount or direction of tilt at all. If you did it right (and I’m sure you did) you will find that the center of buoyancy has migrated in the direction of the tilt. Why is that? It’s because when you tilted your pan you changed the shape of it that was under water (its underwater profile) so that there was a greater volume under water in the direction of the tilt and therefore the force buoyancy was shifted in that direction.

When our ship is lying up straight in the water in perfect trim, if we were to look at a cross section of our hull we would find the center of gravity (G) and the center of buoyancy (B) resting directly on a centerline drawn from the keel straight up through the center of the ship. That’s 7,000 tons of weight (G) exactly opposed by 7,000 tons of buoyancy (B) acting directly in line with each other and in opposite directions.

This illustration shows a hull cross section with the center of gravity and center of buoyancy aligned as equal and opposite forces.

The third point important to our discussion is the metacenter (M). The metacenter is the point at which the upward buoyant force (B) crosses our perpendicular centerline. So, with our ship lying straight with no tilt at all and with the center of buoyancy exerting its force directly upward on the centerline, then the metacenter will be on that line also. But where? How can we find it? The force of buoyancy doesn’t cross the centerline because it’s directly on it.

The way this can be done is to tilt or heel the ship very slightly at a small angle. We know that when we do this the center of buoyancy will move off the centerline slightly in the direction of the tilt. With “B” displaced we can observe where this upward force crosses the perpendicular centerline. That’s our initial metacenter (M). Although it too can migrate under extreme conditions, it’s going to stay put at small angles of heel and that’s all we need to understand the cause of the capsize of Sewol.

These illustrations show how the metacenter is located vertically on the perpendicular centerline.

Now lets see what happens when our ship is caused to heel over from some external force, for example a wave or the centrifugal force created by a turn. As the ship is caused to heel over the part of the hull which is under water changes. Because the force of buoyancy is distributed along the underwater hull, this causes the center of buoyancy to move off the centerline in the direction of the heel and become out of alignment with the center of gravity which stays on the centerline. Because the center of gravity has stayed on the centerline it will appear to swing in a direction opposition the direction of the heel as the ship rotates. The misalignment of these forces, with one pushing straight up through the metacenter and the other pulling straight down will cause what’s called a “moment” in physics; that is, a torque or rotational force. This rotational force will cause the ship to return to the upright position and is called a “righting moment.”

Here is an illustration, again taken from the Handbook of Damage Control.

The strength of this righting moment can be calculated for various angles of heel. The way this is done is to measure the shortest distance from the center of gravity (G) which remains located on our centerline, to the line of upward buoyant force from the displaced center of buoyancy to the metacenter (BM). Finding the shortest distance will always give us a perpendicular or right angle intersection with the line BM and we will call that point of intersection “Z.” The distance measured in feet from G to Z (GZ) is our “righting arm” for that angle of heel. Here’s another illustration.

If we multiply the distance (GZ) times the displacement of our vessel (W) we will have the force of the righting moment denominated in foot-tons. For example, if we assume that at a small angle of heel our center of buoyancy (B) is displaced six inches or one-half foot, we can multiply that times our 7,000 ton displacement and determine that at that particular angle of heel there will be a 3,500 foot-ton force working to restore the vessel to it’s upright position.

In looking at the triangle in the illustration above it is clear that if we keep the angle of heel constant and shorten the metacentric height (GM) by raising the center of gravity (G) toward the metacenter (M), the righting arm (GZ) will get shorter. It is also clear that if we keep the metacentric height (GM) constant and increase the angle of heel, the length of the righting arm (GZ) will increase.

I’ve just described how a righting arm and righting moment can be calculated for one particular metacentric height (GM) and for one particular angle of heel. This can obviously be done for various metacentric heights (GM) and various angles of heel and the inter-relationship between the length of the righting arm(GZ) (and therefore the strength of the righting moment), the angle of heel, and the metacentric height (GM) can be expressed mathematically.

But I did promise not to get too far into the weeds with this, didn’t I? And besides, as you will see, what initiated the Sewol ferry disaster happened at a very small angle of heel. So I’ll ask you to just accept that as the angle of heel increases the righting moment will increase for relatively small angles of heel. It will continue to increase at increasing angles of heel until it reaches a maximum point. Thereafter, as the angle of heel increases the righting moment will decrease until it vanishes at what (with a lack of imagination) is called the “vanishing point.” If the heel is increased even further, the righting moment will become negative. We’ll see what that means next.

NEGATIVE RIGHTING ARM AND MOMENT

Thus far I have described what happens with a properly designed and properly loaded ship when it is acted upon by an external force that causes it to heel. That is, how the buoyant force pushing upward through the metacenter and the force of gravity pulling down create a righting moment; a strong force that works to restore the ship to its upright position.

Now lets consider what happens when the center of gravity (G) is raised to the point that it is higher than the metacenter (M). All of these lines of force still exist but their behavior is very different. The center of buoyancy is still displaced when the ship heels and the center of gravity (G) and the metacenter (M) still sit on the centerline. Gravity still pulls down and buoyancy still pushes up through the metacenter just like before. So what’s different?

What’s different is that as the ship heels the center of gravity, which is now above the metacenter, rotates in the direction of the heel rather than away from it. The forces are still there but they are reversed. Instead of working to restore the ship to an upright position, they now work to continue the rotation of the ship in the same direction as the initial heel. The righting arm and righting moment are now negative.

So how is this negative righting arm manifested when some force such as a wave or wind or the centrifugal force of a turn causes a small heel? Well, the forces are still generated. In our example of a one half foot movement of the center of buoyancy off the centerline and a 7,000 ton displacement we immediately have a 3,500 foot-ton force, but it’s no longer a righting moment, it’s an upsetting moment working to further increase the rotation and cause a capsize.

ANGLE OF LOLL Now remember, that as the heel increases initially so does the moment of force. What you can see in the real world with this set of conditions is a rapid and severe roll resulting from a negative righting moment taking hold after even a very small initiating force (beginning to sound familiar?).

As the angle of heel increases both the center of buoyancy and the center of gravity are now moving in the same direction as the heel. Since the buoyant force continues to operate vertically as it moves off the centerline, it is possible to see the center of buoyancy move back under the center of gravity which remains on the centerline. Remember that by definition, the point where the force of buoyancy crosses the centerline is the metacenter. So in this situation the center of gravity and metacenter would be juxtaposed resulting in a zero righting arm so that the ship will, at least momentarily, stabilizes at a severe angle of rotation. Say, for example, 15̊.

This kind of stabilized rotation is not referred to as a “list,” but as an angle of “loll.” The ship might remain stable at this angle until the forces bringing it about are altered either externally or internally. "List" or "loll", what’s the difference you say? Well, the difference lies in the cause of the rotation and therefore what you need to do to fix it.

A list is caused by an unequal transverse distribution of weight. For example, loading all the lead acid batteries on the port side and the cotton balls on the starboard side. A list then, can be corrected by shifting weight from one side to the other to correct the weight distribution. Move some of the lead acid batteries to the starboard side and some of the cotton balls to the port side until balance is achieved. If you were to do that with a loll, however, you would not bring the ship to an upright position, you would find yourself rotating the ship to a similar angle of loll on the opposite side. Why? Because transverse weight distribution was not the problem to begin with. The problem was that the center of gravity was higher than the metacenter. So to stabilize the ship from an angle of loll you need to lower the center of gravity, not shift weight from one side to the other. You might do that, for example by jettisoning deck cargo and flooding ballast tanks.

THE REGULATORY SCHEME—CLASSIFICATION SOCIETIES

There are more things we’ll need to do before we book our first passengers and take on a load of cargo. For example, we’ll want to comply with all of the regulatory requirements (and there are lots) and we’ll want to insure our ship in case something bad happens.

It’s a good thing we’ve been wise enough to keep some reserve buoyancy and we’ll want to put our Plimsol mark on the ship in the right place to make sure we can see if we have enough. But how much is enough? Can we decide for ourselves what load we can carry and how much freeboard we need? Of course not. “Big brother” does not trust us.

Remember we started this project way back with our cube of steel in order to make some money. And we make money by carrying passengers and cargo. This is the easy part of the math, right? Every extra ton of cargo we carry means more freight (revenue) and every extra passenger is another fare. So there’s the trade-off—more cargo and more passengers means more money (profits). But it also means a deeper draft and less reserve buoyancy (safety).

We know, of course, that you and I would always make the right decision and never sacrifice safety for profits. It’s the rest of the world that we’re worried about, right? Thanks in large part to Mr. Plinsol’s work , there are now standards and regulations set out in international treaties and in the implementing laws and regulations of the signatory nations which set these limits. It’s not up to us to decide how much cargo and how many passengers we can carry or where to put the Plimsol mark on our hull. That will be the business of regulators and classification societies.

Classification societies are non-governmental organizations that set standards, review plans, survey and inspect ships, and issue certificates of compliance. A couple of well know classifications societies are the American Bureau of Shipping (ABS) and Lloyd’s Register (LR). There is also such an organization for Korea known as the Korean Register of Shipping (KR). These classification societies review and inspect many different things to ensure the seaworthiness of a ship. Two of these things important to our discussion that they review and certify are (1) that the vessel has sufficient reserve buoyancy and is not overloaded (by limiting the maximum loaded draft), and (2) ensuring that the vessel has adequate stability for all loading and operating conditions (by issuing approved stability documentation & instructions).

Happily our ship has passed all stages of review and inspection (Joe did a good job) and has been classed to operate on the ocean and carry passengers and cargo. We have been issued a load line certificate or equivalent which sets limits on the load we can carry, tells us our maximum draft and therefore how much reserve buoyancy we are required to have. It also provides us with detailed stability information for various load conditions so that we know what will happen to our righting moment under various load conditions.

I’ve shown you decades old diagrams and mathematical calculations that can be done to determine vessel stability and I’m sure some of you might be wondering “isn’t there an app for that?” Actually there is. Even better than an app, software is now readily available so that all of these calculations are quickly at hand for the vessel’s officers simply by making the necessary inputs into the program.

Alright, so enough of the math and physics. Let’s next look at what happened to the ferry.